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    <title>linmeq</title>
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    <center>Scilab Function</center>
    <div align="right">Last update : 2001</div>
    <p>
      <b>linmeq</b> -  Sylvester and Lyapunov equations solver</p>
    <h3>
      <font color="blue">Calling Sequence</font>
    </h3>
    <dl>
      <dd>
        <tt>[X(,sep)] = linmeq(task,A,(B,)C,flag,trans(,schur))  </tt>
      </dd>
    </dl>
    <h3>
      <font color="blue">Parameters</font>
    </h3>
    <ul>
      <li>
        <tt>
          <b>task</b>
        </tt>: integer option to determine the equation type:<ul>
          <li>
            <tt>
              <b>=1</b>
            </tt>:  solve the Sylvester equation (1a) or (1b);</li>
          <li>
            <tt>
              <b>=2</b>
            </tt>:  solve the Lyapunov equation (2a) or (2b);</li>
          <li>
            <tt>
              <b>=3</b>
            </tt>: solve for the Cholesky factor op(X) the Lyapunov equation (3a) or (3b).</li>
        </ul>
      </li>
      <li>
        <tt>
          <b>A</b>
        </tt>: real matrix</li>
      <li>
        <tt>
          <b>B</b>
        </tt>: real matrix</li>
      <li>
        <tt>
          <b>C</b>
        </tt>: real matrix</li>
      <li>
        <tt>
          <b>flag</b>
        </tt>: (optional) integer vector of length 3 or 2 containing options.<ul>
          <li>
            <tt>
              <b>task</b>
            </tt>= 1 : flag has length 3<ul>
              <li>
                <tt>
                  <b>flag(1)</b>
                </tt>= 0 : solve the continuous-time equation (1a); otherwise, solve the discrete-time equation (1b).</li>
              <li>
                <tt>
                  <b>flag(2)  </b>
                </tt>= 1 : A is (quasi) upper triangular;</li>
              <li>
                <tt>
                  <b>flag(2)  </b>
                </tt>= 2 : A is upper Hessenberg;</li>
              <li>
                <tt>
                  <b>otherwise</b>
                </tt>A is in general form.</li>
              <li>
                <tt>
                  <b>flag(3)  </b>
                </tt>= 1 : B is (quasi) upper triangular;</li>
              <li>
                <tt>
                  <b>flag(3)  </b>
                </tt>= 2 : B is upper Hessenberg;</li>
              <li>
                <tt>
                  <b>otherwise,  </b>
                </tt>B is in general form.</li>
            </ul>
          </li>
          <li>
            <tt>
              <b>task</b>
            </tt>= 2 : flag has length 2<ul>
              <li>
                <tt>
                  <b>flag(1)</b>
                </tt>: if 0 solve continuous-time equation
            (2a), otherwise, solve discrete-time equation (2b).</li>
              <li>
                <tt>
                  <b>flag(2) </b>
                </tt>= 1 : A is (quasi) upper triangular     otherwise, A is in general form.</li>
            </ul>
          </li>
          <li>
            <tt>
              <b>task</b>
            </tt>= 3 : flag has length 2<ul>
              <li>
                <tt>
                  <b>flag(1)  </b>
                </tt>= 0 : solve continuous-time equation (3a); otherwise, solve discrete-time equation (3b).</li>
              <li>
                <tt>
                  <b>flag(2)  </b>
                </tt>= 1 : A is (quasi) upper triangular; otherwise, A is in general form.</li>
            </ul>
          </li>
        </ul>
        <p>
     Default:    flag(1) = 0, flag(2) = 0 (, flag(3) = 0).
    </p>
      </li>
      <li>
        <tt>
          <b>trans</b>
        </tt>: (optional) integer specifying a transposition option.<ul>
          <li>
            <tt>
              <b>=  </b>
            </tt>0 : solve the equations (1) - (3) with op(M) = M.</li>
          <li>
            <tt>
              <b>=  </b>
            </tt>1 : solve the equations (1) - (3) with op(M) = M'.</li>
          <li>
            <tt>
              <b>=  </b>
            </tt>2 : solve the equations (1) with op(A) = A';  op(B) = B;</li>
          <li>
            <tt>
              <b>=  </b>
            </tt>3 : solve the equations (1) with op(A) = A;   op(B) = B'.</li>
        </ul>
        <p>
     Default:    trans = 0.
    </p>
      </li>
      <li>
        <tt>
          <b>schur</b>
        </tt>: (optional) integer specifying whether the Hessenberg-Schur or Schur method should be used. Available for task = 1.<ul>
          <li>
            <tt>
              <b>= 1 : Hessenberg-Schur method (one matrix is reduced</b>
            </tt>to Schur form).</li>
          <li>
            <tt>
              <b>= 2 : Schur method (two matrices are reduced to Schur</b>
            </tt>form).</li>
        </ul>
        <p>
    Default:    schur = 1.
  </p>
      </li>
      <li>
        <tt>
          <b>X</b>
        </tt>:</li>
      <li>
        <tt>
          <b>sep</b>
        </tt>: (optional) estimator of Sep(op(A),-op(A)') for (2.a) or Sepd(A,A') for (2.b).</li>
    </ul>
    <h3>
      <font color="blue">Description</font>
    </h3>
    <p>
    linmeq  function for solving Sylvester and Lyapunov equations
    using SLICOT routines SB04MD, SB04ND, SB04PD, SB04QD, 
    SB04RD, SB03MD, and SB03OD.
  </p>
    <pre>

       [X] = linmeq(1,A,B,C,flag,trans,schur)
   [X,sep] = linmeq(2,A,C,flag,trans)
       [X] = linmeq(2,A,C,flag,trans)
       [X] = linmeq(3,A,C,flag,trans)
   
    </pre>
    <p>
    linmeq solves various Sylvester and Lyapunov matrix equations:
  </p>
    <pre>

        op(A)*X + X*op(B) = C,                           (1a)

        op(A)*X*op(B) + X = C,                           (1b)

        op(A)'*X + X*op(A) = C,                          (2a)

        op(A)'*X*op(A) - X = C,                          (2b)

        op(A)'*(op(X)'*op(X)) + (op(X)'*op(X))*op(A) =
                              -  op(C)'*op(C),           (3a)

        op(A)'*(op(X)'*op(X))*op(A) - op(X)'*op(X) =
                                    - op(C)'*op(C),      (3b)
   
    </pre>
    <p>
    where op(M) = M, or M'.
  </p>
    <h3>
      <font color="blue">Comments</font>
    </h3>
    <dl>
      <dd>
        <li>
          <b>
            <font color="maroon">1.</font>
          </b>For equation (1a) or (1b), when schur = 1, the Hessenberg-Schur
method is used, reducing one matrix to Hessenberg form and the other
one to a real Schur form. Otherwise, both matrices are reduced to real
Schur forms. If one or both matrices are already reduced to
Schur/Hessenberg forms, this could be specified by flag(2) and
flag(3). For general matrices, the Hessenberg-Schur method could be
significantly more efficient than the Schur method.</li>
        <li>
          <b>
            <font color="maroon">2.</font>
          </b>For equation (2a) or (2b), matrix C is assumed symmetric.</li>
        <li>
          <b>
            <font color="maroon">3.</font>
          </b>For equation (3a) or (3b), matrix A must be stable or convergent, respectively.</li>
        <li>
          <b>
            <font color="maroon">4.</font>
          </b>For equation (3a) or (3b), the computed matrix X is the Cholesky factor of the solution, i.e., the real solution is op(X)'*op(X), where X is an upper triangular matrix.</li>
      </dd>
    </dl>
    <h3>
      <font color="blue">Revisions</font>
    </h3>
    <dl>
      <p>
    V. Sima, Katholieke Univ. Leuven, Belgium, May 1999, May, Sep. 2000. V. Sima, University of Bucharest, Romania, May 2000.</p>
    </dl>
    <h3>
      <font color="blue">Examples</font>
    </h3>
    <pre>

//(1a)
n=40;m=30;
A=rand(n,n);C=rand(n,m);B=rand(m,m);
X = linmeq(1,A,B,C);
norm(A*X+X*B-C,1)
//(1b)
flag=[1,0,0]
X = linmeq(1,A,B,C,flag);
norm(A*X*B+X-C,1)
//(2a)
A=rand(n,n);C=rand(A);C=C+C';
X = linmeq(2,A,C);
norm(A'*X + X*A -C,1)
//(2b)
X = linmeq(2,A,C,[1 0]);
norm(A'*X*A -X-C,1)
//(3a)
A=rand(n,n);
A=A-(max(real(spec(A)))+1)*eye(); //shift eigenvalues
C=rand(A);
X=linmeq(3,A,C);
norm(A'*X'*X+X'*X*A +C'*C,1)
//(3b)
A = [-0.02, 0.02,-0.10, 0.02,-0.03, 0.12;
      0.02, 0.14, 0.12,-0.10,-0.02,-0.14;     
     -0.10, 0.12, 0.05, 0.03,-0.04,-0.04;     
      0.02,-0.10, 0.03,-0.06, 0.08, 0.11;      
     -0.03,-0.02,-0.04, 0.08, 0.14,-0.07;   
      0.12,-0.14,-0.04, 0.11,-0.07, 0.04]    

C=rand(A);
X=linmeq(3,A,C,[1 0]);
norm(A'*X'*X*A - X'*X +C'*C,1)
 
  </pre>
    <h3>
      <font color="blue">See Also</font>
    </h3>
    <p>
      <a href="../linear/sylv.htm">
        <tt>
          <b>sylv</b>
        </tt>
      </a>,&nbsp;&nbsp;<a href="../linear/lyap.htm">
        <tt>
          <b>lyap</b>
        </tt>
      </a>,&nbsp;&nbsp;</p>
    <h3>
      <font color="blue">Author</font>
    </h3>
    <p>H. Xu, TU Chemnitz, FR Germany, Dec. 1998.  </p>
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